superfactorial

In mathematics, and more specifically number theory, the superfactorial of a positive integer $n$ is the product of the first $n$ factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

The $n$ th superfactorial $\mathit{sf}(n)$ may be defined as:{{r|oeis}}

$\begin{align}
\mathit{sf}(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_{i=1}^{n} i! = n!\cdot\mathit{sf}(n-1)\\
&= 1^n \cdot 2^{n-1} \cdot \cdots n = \prod_{i=1}^{n} i^{n+1-i}.\\
\end{align}$

Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with $\mathit{sf}(0)=1$ , is:{{r|oeis}}

{{bi|left=1.6|1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... {{OEIS|A000178}}}}

Properties

Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function.{{r|barnes}}

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $p$ is an odd prime number

$\mathit{sf}(p-1)\equiv(p-1)!!\pmod{p},$

where $!!$ is the notation for the double factorial.{{r|wilson}}

For every integer $k$ , the number $\mathit{sf}(4k)/(2k)!$ is a square number. This may be expressed as stating that, in the formula for $\mathit{sf}(4k)$ as a product of factorials, omitting one of the factorials (the middle one, $(2k)!$ ) results in a square product.{{r|square}} Additionally, if any $n+1$ integers are given, the product of their pairwise differences is always a multiple of $\mathit{sf}(n)$ , and equals the superfactorial when the given numbers are consecutive.{{r|oeis}}

References

{{reflist|refs=

{{citation

| last = Barnes | first = E. W. | author-link = Ernest Barnes

| jfm = 30.0389.02

| journal = The Quarterly Journal of Pure and Applied Mathematics

| pages = 264–314

| title = The theory of the {{mvar|G}}-function

| url = https://gdz.sub.uni-goettingen.de/id/PPN600494829_0031?tify={%22pages%22:[268],%22view%22:%22toc%22}

| volume = 31

| year = 1900}}

{{cite OEIS|1=A000178 |2=Superfactorials: product of first n factorials|mode=cs2}}

{{citation

| last1 = White | first1 = D.

| last2 = Anderson | first2 = M.

| date = October 2020

| doi = 10.1080/10511970.2020.1809039

| issue = 10

| journal = PRIMUS

| pages = 1038–1051

| title = Using a superfactorial problem to provide extended problem-solving experiences

| volume = 31| s2cid = 225372700

}}

{{citation

| last1 = Aebi | first1 = Christian

| last2 = Cairns | first2 = Grant

| doi = 10.4169/amer.math.monthly.122.5.433

| issue = 5

| journal = The American Mathematical Monthly

| jstor = 10.4169/amer.math.monthly.122.5.433

| mr = 3352802

| pages = 433–443

| title = Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials

| volume = 122

| year = 2015| s2cid = 207521192

}}

External links

{{MathWorld|id=Superfactorial|title=Superfactorial|mode=cs2}}

Category:Integer sequences

Category:Factorial and binomial topics